We refer the interested reader to [1]}, [2]} for more details about this notion and in particular about the descendant
condition. Here, we just observe that the notions of hypertree width and generalized hypertree width are true generalizations of acyclicity, as
the acyclic hypergraphs are precisely those hypergraphs having hypertree width and generalized hypertree width one. In particular, the classes
of CSP instances having bounded (generalized) hypertree width have the same desirable computational properties as
acyclic CSPs [3]}. Indeed, from a CSP instance \(\mathcal {I}=\langle {\it Var},U,\)  and a (generalized) hypertree decomposition \(H\!D\)  of
\({\cal H}(\mathcal {I})\)  of width \(k\) , we may build an acyclic CSP instance \(\mathcal {I}^{\prime }=\langle {\it Var},U,\rangle \)  with the same solutions as \(\mathcal {I}\) . The overall cost of
deciding whether \(\mathcal {I}\)  is satisfiable is in this case \(O((m-1) \times r_{max}^k \times \log r_{max}^k)\) , where \(r_{max}\)  denotes the size of the
largest constraint relation and \(m\)  is the number of vertices of the decomposition tree, with \(m\le |{\it Var}|\)  (in that we may always find
decompositions in a suitable normal form without redundancies, so that the number of vertices in the tree cannot exceed the number of variables
of the given instance). To be complete, if the input consists of \(\mathcal {I}\)  only, we have to compute the decomposition, too. This can be done with a
guaranteed polynomial-time upper bound in the case of hypertree decompositions [1]}.
